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7 SPACES OF CONTINUOUS MULTILINEAR MAPPINGS 109

which proves that x->u_x (which is obviously linear) is continuous, and
u-»ui$ an isometry of JS?(E, F; G) into & (E; 56 (F; G)). Finally u-+u is
surjective, for if ve^(E; £?(F; G)), then u: (x, y)-^ (v(x))(y) is obviously
bilinear, and as ||(»(*))GOII < K*)ll • Ibll <IN ' IMI • Ibll by (5.7.4), u is
continuous, and t?(x) = u_x, which ends the proof.

By induction on w, it follows that «S?(E_l9 E₂, ..., E_M; F) can be naturally
identified (with conservation of the norm) to

PROBLEMS

1. Let E be the space (c₀) of Banach, defined in Section 5.3, Problem 5; we keep the nota-
tions of that problem. Let u be a continuous linear mapping of E into R; if u(e_n) = rj_n ,
show that the series £77,, is absolutely convergent, and that, in the Banach space E' =

co n

JSf(E; R), \\u\\ = £ \f)_n\ (apply (5.5.1 ) for suitable values of x e E). Conversely, for any
absolutely convergent series (rj_n) of real numbers, there is one and only one continuous

linear mapping u of E into R such that u(e_n) = r)_n for every n\ and if x = ]T £_ne_n e E,

oo n = 0

then u(x) = V ri_ng_n (the space E', with the norm defined above, is the "space 7¹ " of

n = 0

Banach).

(b) As a vector space (without a norm) E' can be considered as a subspace of E; show
that the norm on E' is strictly finer (Section 5.6) than the restriction to E' of the norm
ofE.

(c) Show that the space E" = J5?(E' ; R) of the continuous linear mappings of E' into R
can be identified with the space of all bounded sequences x — (£„) of real numbers, with
norm \\x\\ = sup |£_n| ("space /°° " of Banach; use the same method as in (a)). E can be

considered as a closed subspace of E".

(d) In the space E', let P be the subset of all absolutely convergent series u = (rj_n)
with terms 77,, ^ 0; any element of E^x can be written u — v, where both u and v are in P;
yet show that the interior of the set P is empty.

2. (a) Let E be the space (c₀) of Banach, and let U be a continuous linear mapping of E

into itself. With the notations of Problem 1, let £7 (<?_n) = £ a_mne_m ; show that: (1)

m = 0

oo oo

lim a_mn — 0; (2) the series ]£ |a_mn| is convergent for every m; (3) sup 2 l^a"»»l is finite.

m-»oo »»0 m ns=0

(Same method as in Problem 1 (a).) Prove the converse, and show that the Banach space
; E) can be identified with the space of double sequences (7= (a_wn) satisfying the

preceding conditions, with the norm \\U\\ = sup £ \<x_mn\.

m n = Q

(b) Let E' be the space I¹ of Banach (Problem 1). Show similarly that the Banach
space JSf(E'; E') can be identified with the space of double sequences C/— (a_mn) such

co co

that: (1) the series ]£ |a_nm| is convergent for every n; (2) sup ]£ \a_mn\ is finite; the

m=sO » m = 0

norm is then equal to ||C/|| = sup 2 l#mn|-



	7 SPACES OF CONTINUOUS MULTILINEAR MAPPINGS 109 which proves that x->u_x (which is obviously linear) is continuous, and u-»ui$ an isometry of JS?(E, F; G) into & (E; 56 (F; G)). Finally u-+u is surjective, for if ve^(E; £?(F; G)), then u: (x, y)-^ (v(x))(y) is obviously bilinear, and as \|\|(»())GOII < K)ll • Ibll <IN ' IMI • Ibll by (5.7.4), u is continuous, and t?(x) = u_x, which ends the proof. By induction on w, it follows that «S?(E_l9 E₂, ..., E_M; F) can be naturally identified (with conservation of the norm) to

	PROBLEMS 1. Let E be the space (c₀) of Banach, defined in Section 5.3, Problem 5; we keep the nota- tions of that problem. Let u be a continuous linear mapping of E into R; if u(e_n) = rj_n , show that the series £77,, is absolutely convergent, and that, in the Banach space E' = co n JSf(E; R), \\u\\ = £ \f)_n\ (apply (5.5.1 ) for suitable values of x e E). Conversely, for any absolutely convergent series (rj_n) of real numbers, there is one and only one continuous oo linear mapping u of E into R such that u(e_n) = r)_n for every n\ and if x = ]T £_ne_n e E, oo n = 0 then u(x) = V ri_ng_n (the space E', with the norm defined above, is the "space 7¹ " of n = 0 Banach). (b) As a vector space (without a norm) E' can be considered as a subspace of E; show that the norm on E' is strictly finer (Section 5.6) than the restriction to E' of the norm ofE. (c) Show that the space E" = J5?(E' ; R) of the continuous linear mappings of E' into R can be identified with the space of all bounded sequences x — (£„) of real numbers, with norm \\x\\ = sup \|£_n\| ("space /°° " of Banach; use the same method as in (a)). E can be n considered as a closed subspace of E". (d) In the space E', let P be the subset of all absolutely convergent series u = (rj_n) with terms 77,, ^ 0; any element of E^x can be written u — v, where both u and v are in P; yet show that the interior of the set P is empty. 2. (a) Let E be the space (c₀) of Banach, and let U be a continuous linear mapping of E oo into itself. With the notations of Problem 1, let £7 (<?_n) = £ a_mne_m ; show that: (1) m = 0 oo oo lim a_mn — 0; (2) the series ]£ \|a_mn\| is convergent for every m; (3) sup 2 l^a"»»l is finite. m-»oo »»0 m ns=0 (Same method as in Problem 1 (a).) Prove the converse, and show that the Banach space ; E) can be identified with the space of double sequences (7= (a_wn) satisfying the

	preceding conditions, with the norm \\U\\ = sup £ \<x_mn\. m n = Q (b) Let E' be the space I¹ of Banach (Problem 1). Show similarly that the Banach space JSf(E'; E') can be identified with the space of double sequences C/— (a_mn) such co co that: (1) the series ]£ \|a_nm\| is convergent for every n; (2) sup ]£ \a_mn\ is finite; the m=sO » m = 0 oo norm is then equal to \|\|C/\|\| = sup 2 l#mn\|-